Construction of nonseparable orthonormal compactly supported wavelet bases for L2(Rd)

Suppose M and N are two r × r and s × s dilation matrices, respectively. Let ?? _M and ?? _N represent the complete sets of representatives of distinct cosets of the quotient groups M ^?T ? ^r /? ^r and N ^?T? ^s /? ^s , respectively. Two methods for constructing nonseparable ??-filter banks from M-filter banks and N-filter banks are presented, where ?? is a (r + s) × (r + s) dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups ?? ^?T ? ^r+s /? ^r+s are ??_?? = {[?? _h ^T , ?? _q ^T ] ^T : ?? _h ?? ?? _M , ?? _q ?? ?? _N }. Specially, ?? can be $\left[ {\begin{array}{*{20}c} {M\Theta } \\ {0N} \\ \end{array} } \right] $ , where ?? is a r × s integer matrix with M ^?1?? being also an integer matrix. Moreover, if the constructed filter bank satisfies Lawton??s condition, which can be easy to verify, then it generates an orthonormal nonseparable ??-wavelet basis for L ²(? ^r+s ). Properties, including Lawton??s condition, vanishing moments and regularity of the new ??-filter banks or new ??-wavelet basis are discussed then. Finally, a class of nonseparable ??-wavelet basis for L ²(? ^r+1) are constructed and three other examples are given to illustrate the results. In particular, when M = N = 2, all results obtained in this paper appeared in [1].